Q 9. For a reaction, specific rate constant at 283 K is 2.5 × 10-6 L mol-1s-1 and at 293 K is 4.25 × 10-5 L mol-1s-1. Compute the energy of activation of the reaction.

To determine the energy of activation (Ea) for the reaction, we can use the Arrhenius equation, which relates the rate constant (k) to temperature (T) and the activation energy. The equation is expressed as:

k = A e^(-Ea/RT)

Where:

• k = rate constant
• A = pre-exponential factor
• R = universal gas constant (8.314 J/mol·K)
• T = temperature in Kelvin

To find the activation energy, we can rearrange the Arrhenius equation into a linear form. Taking the natural logarithm of both sides gives us:

ln(k) = ln(A) - (Ea/RT)

This can be rearranged to:

ln(k2/k1) = -Ea/R (1/T2 - 1/T1)

Here, k1 and k2 are the rate constants at temperatures T1 and T2, respectively. We have:

• k1 = 2.5 × 10^-6 L mol^-1 s^-1 at T1 = 283 K
• k2 = 4.25 × 10^-5 L mol^-1 s^-1 at T2 = 293 K

Now, we can plug in the values into the equation:

ln(k2/k1) = ln(4.25 × 10^-5 / 2.5 × 10^-6)

Calculating the ratio:

k2/k1 = 4.25 × 10^-5 / 2.5 × 10^-6 = 17

Now, taking the natural logarithm:

ln(17) ≈ 2.833

Next, we need to calculate the temperature difference:

1/T2 - 1/T1 = 1/293 - 1/283

Calculating each term:

1/293 ≈ 0.003414 and 1/283 ≈ 0.003531

Thus:

1/T2 - 1/T1 ≈ 0.003414 - 0.003531 = -0.000117

Now we can substitute these values back into the rearranged Arrhenius equation:

2.833 = -Ea/(8.314)(-0.000117)

Rearranging to solve for Ea gives:

Ea = 2.833 × 8.314 × 0.000117

Calculating this yields:

Ea ≈ 0.000334 J/mol

Finally, converting this to kJ/mol (since activation energy is often expressed in kJ/mol):

Ea ≈ 334 J/mol = 0.334 kJ/mol

In summary, the energy of activation for the reaction is approximately 0.334 kJ/mol. This value indicates the minimum energy required for the reactants to undergo the reaction, providing insight into the reaction's kinetics and temperature dependence.